5e, more than any other addition (with the possible exception of 4e), lessons the "we gotta have a cleric" requirement. Between Bards, Druids, Paladins, Rangers and such all having access to healing spells to a greater or lesser degree, and the proliferation of self recover via features like Second Wind, Druidic polymorphing, short rests and other abilities, there just is not as great a need for a cleric healer as there was once upon a time. This frees up the cleric to be more of the support-protect-uplift-debuff class without having to spend so much of their power on healing.
Besides, I agree with the poser above in that I have found that excessive healing tends to lend to a more dragged out, up-down-up type of play over the last few editions that I personally find less satisfying. One can still be the classic cleric and provide support and protection, and be particularly useful in an undead themed game/adventure, without having to be a heal bot.
So, back to the original question: Yes, you can not have a cleric (or clerics in general) in any particular campaign and have it work well. Depending on the party make up and adventure type, you could hardly notice the loss.